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<!DOCTYPE html>
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<hr>
<h2 class="sticky" id="Notes">Notes</h2>
<h4>Preface</h4>
<p> In the fall of 2022 it was my great pleasure to see and hear the orcas
of J pod at Lime Kiln State Park with David Neiwert (see bibliography)
and his son Devin.</p>
<h4>Introduction</h4>
<p>Einstein's visit with Henry Ford is entirely a product of my imagination.
Had Ford asked Einstein in 1935 for a simple thought experiment to convey
the meaning and essence of relativity, Einstein would have been more
likely to have taken Ford to look down from the top of the university's
highest building and then to have explained that the happiest thought of
his career was that a man falling from a rooftop does not feel any of the
effects of gravity (until he strikes ground, of course). We will analyze
and criticize Einstein's <q>happiest thought</q> in Chapter Twenty-Nine.</p>
<h4>Fit the First: <i>On Fugues</i></h4>
<p>It has been said that imitation is the sincerest form of flattery, and
you will find much of that here. This vignette is in equal parts a
shameless imitation of, an adoring homage to, and an unfair satire of,
Douglas Hofstadter's famous tome <cite>Gödel, Escher,
Bach</cite>. It is therefore equally self-referential, and any facetious
criticism of his work implicit in what I have written here is at least as
applicable to my own. Anyone even passingly familiar with Hofstadter's
book is likely to notice at several points in this one where I have taken
cues and inspiration from <cite>GEB</cite>, though I must confess that even
after decades of admiring it I — like Achilles — have yet to reach
its finish line. In Hofstadter's honor, I have given the tortoise the
feminine gender, a decision he wished to have come to earlier than he did.
Lewis Carroll, from whom Hofstadter took his own inspiration, gave the
tortoise no gender at all; Hofstadter's tortoise became feminine in some
of his book's foreign editions.</p>
<p>The use of the word <q>fit</q> to title subdivisions of a longer work
originates with Lewis Carroll's poem <cite>The Hunting of the Snark</cite>
("An Agony in Eight Fits") This convention was imitated by the radio
series <cite>The Hitchhiker's Guide to the Galaxy</cite>. Speaking of the <cite>Guide</cite>,
the tablet the tortoise reads from is in some respects similar. Douglas
Adams anticipated Wikipedia in some ways, but allowed the entire
enterprise to fall under the control of the Vogons ("a horde of
bureaucrats who'd be perfectly happy to destroy this planet and everyone
on it"). The tortoise having found the tablet discarded by the roadside
stems from my memory of Arthur Dent having thrown his copy of the Guide
into a river on prehistoric Earth. Achilles asking the tortoise what she
is reading is intended as an imitation to Hamlet's dialogue with Polonius,
which I re-imagine later in this book.
<!-- https://www.gutenberg.org/cache/epub/13/pg13-images.html--></p>
<h4> Chapter One: Mapping Space</h4>
<p><i>Recommended resources</i>:</p>
<ul>
<li>This <a href="https://www.youtube.com/watch?v=pAlq9fFwtus" target="_blank">video
</a>by the Khan Academy on Cartestian coordinates.</li>
<li>Ferris' <cite>Coming of Age in the Milky Way</cite> (see <a href="#Bibliography">bibliography</a>),
particularly for its discussion of how mankind's understanding of space
has evolved over time. </li>
</ul>
<h4>Chapter Two: Mapping Time</h4>
<p>Long before Magellan's time and hundreds of years before the
implementation of the International Date Line, the Arab geographer <a href="https://en.wikipedia.org/wiki/Abulfeda">Abulfeda</a>
(1273-1331) made the earliest known prediction that those who
circumnavigated the globe would accumulate a one-day offset in their
reckoning of time, depending on which direction they traveled. Alas,
Magellan and Mr. Fogg were not aware.</p>
<p>Figure 2-2 is described as showing the earth's rotation <q>at equinox</q>
because it is only during these two times of year that the north and south
poles would be on the line between day and night, or more precisely,
between sunlight and shadow. Due to the tilt of the earth relative to its
orbit around the sun, the poles remain in the earth's shadow for six
months at a time.</p>
<h4>Fit the Second: <i>Circus Canon</i></h4>
<p>In the early 1930s, Einstein did come to Oxford as a guest lecturer and
stayed in the rooms once occupied by Lewis Carroll, whose other
professional identity was the mathematician Charles Dodgson (see
Robinson's <cite>Einstein On The Run</cite>, p. 173).</p>
<p>Does anyone here remember usenet? <br>
<!--(Robinson, p. 173) Einstein stayed in Carroll's room. First in May 1931 to give three lecturers. Came to Oxford again May '32, then again May-Jun 1933--></p>
<h4>Chapter Eight: Mechanical Perspective (I)</h4>
<p>"<i>Even now, physicists occasionally stumble when relating projectile
motion below to planetary orbits above.</i>" See, for instance, Lee
Smolin's <cite>Time Reborn</cite> (page 21): "Ellipses trace the planetary
orbits and parabolas trace the paths of falling bodies on Earth." This
fallacy is discussed at length in Chapter Sixteen.</p>
<p><i>Recommended resources</i>:</p>
<ul>
<li><a href="https://www.3blue1brown.com/lessons/vectors" target="_blank">https://www.3blue1brown.com/lessons/vectors</a></li>
</ul>
<h4>Chapter Nine: Mapping Forces</h4>
<p>I wanted to mention Benjamin Franklin in connection with my comments on
static electricity, but I didn't see a place where it wouldn't have
interrupted the flow. Franklin is so well-remembered as a statesman, even
I had started to forget his work as a scientist.</p>
<h4>Chapter Eleven: Math Vocabulary (I)</h4>
<p>"<i>Calculus is thought to have been an invention of necessity for
Newton, though Gottfried Leibniz arrived at many of the same ideas
independently around the same time.</i>" For further information on the
history of the development of calculus, see <a href="https://www.quora.com/Is-it-true-that-calculus-was-invented-before-Leibniz-and-Newton-in-India">https://www.quora.com/Is-it-true-that-calculus-was-invented-before-Leibniz-and-Newton-in-India</a><br>
</p>
<p>"<i>the answer involves calculus (which really isn't that difficult once
you throw away the tedious first third of your textbook dealing with
limits and the fundamental theorem of calculus)</i>" I think it's poor
practice that a typical calculus course subjects students to this material
before getting to the practical applications of calculus. Yes, show us <em>why</em>
calculus works, but only if we care after having been shown <em>how</em>.</p>
<h4>Chapter Twelve: Electromagnetism (I)</h4>
<p>One of the many ideas that caught my eye but didn't make it into the main
text was something in Wolfon and Pasachoff's <cite>Physics for Scientists and
Engineers</cite> (their Figure 18-32b). A ball with angular speed <b>ω</b>
and velocity <b>v</b> through a mass of air experiences a force
<b>ω</b>x<b>v</b> which is analogous to q<b>v</b>
x <b>B</b>. I wouldn't have expected aerodynamics and
electromagnetism to have that math in common, but there we are.</p>
<h4>Chapter Fourteen: Clockwork in Reverse</h4>
<p>Here is the relevant excerpt from Carroll's <cite>Sylvie and Bruno
Concluded</cite> (1893):</p>
<blockquote>
<p><q class="dialog">They run their railway-trains without any engines nothing is needed
but machinery to stop them with. Is that wonderful enough,Miladi?</q></p>
<p><q class="dialog">But where does the force come from?</q> I ventured to ask.</p>
<p>Mein Herr turned quickly round, to look at the new speaker. Then he
took off his spectacles, and polished them, and looked at me again, in
evident bewilderment. I could see he was thinking - as indeed I was also
- that we must have met before.</p>
<p><q class="dialog">They use the force of <em>gravity</em></q> he said. <q class="dialog">It is a force known
also in <em>your</em> country, I believe?</q></p>
<p><q class="dialog">But that would need a railway going <em>down-hill,</em></q> the Earl
remarked. <q class="dialog">You ca'n't have <em>all</em> your railways going
down-hill?</q></p>
<p> <q class="dialog">They <em>all</em> do,</q> said Mein Herr.</p>
<p><q class="dialog">Not from <em>both</em> ends?</q></p>
<p><q class="dialog">From <em>both</em> ends.</q></p>
<p><q class="dialog">Then I give it up!</q> said the Earl.</p>
<p><q class="dialog">Can you explain the process?</q> said Lady Muriel. <q class="dialog">Without using that
language, that I ca'n't speak fluently?</q></p>
<p><q class="dialog">Easily,</q> said Mein Herr. <q class="dialog">Each railway is in a long tunnel, perfectly
straight: so of course the <em>middle</em> of it is nearer the centre
of the globe than the two ends: so every train runs half-way <em>down</em>-hill,
and that gives it force enough to run the <em>other</em> half <em>up</em>-hill.</q>
</p>
<p><q class="dialog">Thank you. I understand that perfectly,</q> said Lady Muriel. <q class="dialog">
But the velocity, in the <em>middle</em> of the tunnel, must be something <em>fearful</em>! </q>
</p> </blockquote>
<h4></h4>
<p> See also <a href="https://en.wikipedia.org/wiki/Gravity_train#Origin_of_the_concept%20%20"
target="_blank">https://en.wikipedia.org/wiki/Gravity_train#Origin_of_the_concept.
<br></a></p>
<p>As an amusing aside, the opposite points of the earth which such a train
might connect are called <em>antipodes</em>. In a much earlier novel,
Carroll's Alice mispronounces the word as <q>antipathies.</q></p>
<p><i>Recommended resources</i>:</p>
<ul>
<li><a href="https://www.3blue1brown.com/lessons/differential-equations" target="_blank">
https://www.3blue1brown.com/lessons/differential-equations</a> A great
presentation on phase space and phase flow.</li>
</ul>
<h4>Fit the Fourth: The Arrow of Time (I)</h4>
<h4><i>Recommended resources</i>: </h4>
<ul>
<li>The Stanford Encyclopedia of Philosophy on Zeno's paradoxes:
<a href="https://plato.stanford.edu/entries/paradox-zeno/"
target="_blank" data-saferedirecturl="https://www.google.com/url?q=https://plato.stanford.edu/entries/paradox-zeno/&source=gmail&ust=1677703953705000&usg=AOvVaw393kC6m43AvypTT4c3Qi4x">
https://plato.stanford.edu/<wbr>entries/paradox-zeno/</a></li>
</ul>
<h4>Chapter <i>i</i>: Imaginary Math</h4>
<p>"Potentially losing some information is a fundamental consequence of
differentiation." The initial values are lost; the indefinite integral
suggests that you can just make one up when you go in reverse, because any
starting value would be valid.</p>
<p>Thanks to <q>Mark44</q> in Washington state for setting me straight when I
couldn't figure out how I was not getting a calculation right (I started
with the wrong formula).
<a href="https://www.physicsforums.com/threads/differentiating-euler-formula-vs-multiplying-by-i.1048002/"
target="_blank">https://www.physicsforums.com/threads/<wbr>differentiating-euler-formula-vs-multiplying-by-i.1048002/</a></p>
<p><i>Recommended resources:</i></p>
<ul>
<li><a href="https://www.3blue1brown.com/lessons/eulers-formula-via-group-theory"
target="_blank">https://www.3blue1brown.com/lessons/eulers-formula-via-group-theory</a>
This is a must-see visualization of how addition and multiplication work
on the real number line and in the complex plane.
<!-- sliding (Additive reals) stretching (multiplicative reals). 2D sliding (additive complex) and stretch/rotation (multiplicative complex) -->
</li>
<li>Penrose, <cite>The Road to Reality</cite>, chapter four. The history of
complex numbers, including cubic solutions.</li>
<li><a href="https://www.3blue1brown.com/lessons/groups-and-monsters" target="_blank">https://www.3blue1brown.com/<wbr>lessons/<wbr>groups-and-monsters</a>
The cubic formula is discussed at about five minutes in.</li>
<li>Veritasium on how imaginary numbers were invented (including the cubic
formula): <a href="%20https://www.youtube.com/watch?v=cUzklzVXJwo" target="_blank">
https://www.youtube.com/watch?v=cUzklzVXJwo</a></li>
<li>On Lewis Carroll's supposed satire of 19th-century mathematics: <a href="https://www.newscientist.com/article/mg20427391-600-alices-adventures-in-algebra-wonderland-solved/"
target="_blank">https://www.newscientist.com/article/<wbr>mg20427391-600-alices-adventures-in-algebra-wonderland-solved/</a></li>
<li><a href="https://www.intmath.com/complex-numbers/9-impedance-phase-angle.php"
target="_blank">https://www.intmath.com/complex-numbers/<wbr>9-impedance-phase-angle.php</a>
Finding the phase angle in an AC circuit. See also <a href="https://www.intmath.com/complex-numbers/6-products-quotients.php" target="_blank">https://www.intmath.com/complex-numbers/6-products-quotients.php</a>
</li>
<li><a href="https://en.wikipedia.org/wiki/Electrical_impedance#Complex_voltage_and_current"
target="_blank">https://en.wikipedia.org/wiki/Electrical_impedance#Complex_voltage_and_current</a>
Complex voltage calculations</li>
<li><a href="https://en.wikipedia.org/wiki/Phasor" target="_blank">https://en.wikipedia.org/wiki/Phasor</a></li>
</ul>
<h4>Chapter Seventeen: Mapping the Earth (II)</h4>
<p>Poincaré's description of a world where changes in size are driven by
temperature calls to mind Alice's ability in Wonderland to shrink by
fanning herself. When Alice is told to <q>keep her temper,</q> the more archaic
second meaning is that she must take care that her changes in size all
happen in proportion rather than, for instance, growing a serpent-like
neck.</p>
<p>I had originally intended to show the metric tensor and the calculation
of distance with covariant vector components, but after realizing I had
gotten it wrong on my first attempt I decided that the whole affair was
better suited for the end notes. What I had forgotten to do was to include
the dot products of the dual bases with themselves.</p>
<p>Since the angle θ between the <em>covariant</em> components <b>v</b><sub>1</sub>
and <b>v</b><sub>2</sub> (and between the dual basis vectors <b>e</b><sup>1</sup> and <b>e</b><sup>2</sup>)
is π (180 degrees) minus the angle φ <b></b>(Figure 17-32)
between the two basis vectors <b>e</b><sub>1</sub>
and <b>e</b><sub>2</sub>, we could rewrite this as :
</p>
<blockquote class="equation">
<p> <i>v</i><sup>2</sup> = <b>(v</b><sub>1</sub>)<b></b><sup>2</sup>
+ (<b>v</b><sub>2</sub>)<b></b><sup>2</sup> -
2( <b>v</b><sub>1</sub> <b>v</b><sub>2</sub>cos(π-
φ )) = = 1<b>v</b><sub>1</sub><b>v</b><sub>1</sub> -
1<b>v</b><sub>1</sub><b>v</b><sub>2</sub>cos(π- φ
) - 1<b>v</b><sub>2</sub><b>v</b><sub>1</sub>cos(π-
φ ) + 1<b>v</b><sub>2</sub><b>v</b><sub>2</sub></p>
<p><b>Equation 17-3.</b> <span class="equation_description">The length of a vector in terms of its
covariant components and the angle between them and between the dual
basis vectors.</span> </p>
</blockquote>
<p>where π- φ is the angle between the two covariant components of <b>v</b>.
</p>
<p>In terms of a metric tensor, this would be as follows in Figure 17-35,
remembering that the elements of this tensor are indexed as shown in
Figure 17-36:</p>
<blockquote>
<p><img class="Figure" loading="lazy" src="images/tensor7.png" style="max-height: 100px;">
<img class="Figure" loading="lazy" src="images/t8neg2.jpg" style="max-height: 100px;">
</p>
<p> <b>Figure 17-35</b>. A metric tensor for use with
contravariant (dual) basis vectors and covariant vector components in a
system with non-orthogonal coordinate axes. <b>
Figure 17-36</b>. The index key for the metric tensor in Figure
17-35.</p>
</blockquote>
<p><i>Recommended resources:</i></p>
<ul>
<li>More on hyperbolic geometry in Penrose, <cite>The Road
to Reality</cite>, chapter two. Also O'Shea's detailed history in <cite>The
Poincaré Conjecture</cite>.</li>
<li>"The meaning of the metric tensor" <a href="https://www.youtube.com/watch?v=Dn0ZZRVuJcU"
target="_blank"> https://www.youtube.com/watch?v=Dn0ZZRVuJcU</a> The
most visual and thorough presentation I have yet seen on the
subject.<!--15:00 three terms in the metric for the 3 lengths of a cube, 3 more terms for the angles between faces-->
</li>
</ul>
<p></p>
<br>
<h4>Chapter Eighteen: The Arrow of Time (II)</h4>
<p><i>Recommended resources:</i></p>
<ul>
<li>The life cycles of stars: <a href="https://imagine.gsfc.nasa.gov/educators/lifecycles/LC_main3.html"
target="_blank">https://imagine.gsfc.nasa.gov/educators/lifecycles/LC_main3.html</a></li>
</ul>
<h4>Chapter Nineteen: Math Vocabulary (II)</h4>
<p><i>Recommended resources:</i></p>
<ul>
<li><a href="https://www.3blue1brown.com/lessons/span%20" target="_blank">https://www.3blue1brown.com/lessons/span</a>
Span, basis, linear combination, redundancy/linear dependence</li>
<li><a href="https://www.3blue1brown.com/lessons/linear-transformations" target="_blank">https://www.3blue1brown.com/lessons/linear-transformations</a>
Linear transformations explained beautifully: the
coefficient columns are a transformation matrix telling us columnwise
where to move the basis vectors (just watch the video).</li>
<li><a href="https://www.3blue1brown.com/lessons/determinant" target="_blank">https://www.3blue1brown.com/lessons/determinant</a>
See the determinant diagram around 8:43.</li>
<li><a href="https://www.3blue1brown.com/lessons/inverse-matrices" target="_blank">https://www.3blue1brown.com/lessons/inverse-matrices</a>
Systems of linear equations and solving them; invertibility</li>
</ul>
<h4>Fit the Fifth: <i>At the Non Sequitur</i></h4>
<p>To those readers wondering what on earth this chapter was all about, I
thank you for indulging me in a bit of fun. This was the
first chapter of the book written largely — if not solely — for the
purpose of entertainment (preferably, yours as well as mine); others
followed, eventually to become this book's <q>fits.</q> It all began with the
idea of Hamlet's tribute to Einstein. Several other aspects of this scene
seemed to develop together, and I decided to lean into the obvious
anachronism of its premise; it presented itself as a humorous way to
precede a more serious discussion of time (and a certain loss of
sequentiality) in the next chapter. I decided to name the inn the Non
Sequitur (Latin for <q>it does not follow</q>), and to play with the comedic <i>non
sequitur</i> in the tradition of two of the great plays whose
characters I have borrowed and placed in this setting.</p>
<p>Alice's tutor would presumably have been Lewis Carroll himself, who
taught mathematics at Oxford under his real name, Charles Dodgson. I have
cast the former Oxford-area furniture dealer Theophilus Carter (a.k.a. the
Mad Hatter) as the owner of this restaurant. See page 69 of <cite>The
Annotated Alice</cite> for more amusing background on the historical and
fictional Mad Hatters.</p>
<p>The inn's relationship with time is not unlike that of the Hatter's Mad
Tea-Party, but perhaps more like the concept of Douglas Adams' Milliways,
<cite>The Restaurant at the End of the Universe</cite> where people "of all
ages" might mingle, and where it is always the same time. Adams borrowed
from Carroll many times; compare his "If you've done 6 impossible things
this morning, why not round it off with breakfast at Milliways" with this
snippet of dialogue from <cite>Through the Looking-Glass</cite>:</p>
<blockquote>
<p>Alice laughed. <q>There's no use trying,</q> she said: <q>one can't believe
impossible things.</q></p>
<p><q>I daresay you haven't had much practice,</q> said the Queen. <q>When I was
your age, I always did it for half-an-hour a day. Why, sometimes I've
believed as many as six impossible things before breakfast.</q></p>
</blockquote>
<p>I only wrote the dialogue around the reception desk in order to smooth
the way for my Edgar Allan Poe reference (which also pointed to the
strange cyclical time of Milliways and the Tea Party); but having done
that, I decided that the hostess ought to give her responses subtly out of
sequence in her dialogue with Alice. Perhaps she is related to the White
Queen?</p>
<p>The philosophers song was written by Eric Idle of Monty Python (<a href="https://www.youtube.com/watch?v=l9SqQNgDrgg" target="_blank">https://www.youtube.com/watch?v=l9SqQNgDrgg</a>). In
regards to the posters on the wall, my non-exhaustive survey of time
travel in cinema finds that the stand-alone movies (those not having
sequels or prequels to speak of) often feature strange loops (which I note
with a nod in the direction of Douglas Hofstadter).</p>
<p>Some of the humor just wrote itself. I didn't have his paradoxes in mind
when I wrote Zeno's comment about not yet having managed to get to the
Restaurant at the End of the Universe; I hope that line strikes others as
funny as it did me.</p>
<p>Alas, something instructive did find its way into this chapter, but I
only allowed it because I found it too amusing. Rosencrantz's retort to
Guildenstern reflects a key concept in this book, indeed one that relates
to its title: that we are not always justified in asserting that something
has or has not yet taken place somewhere else. This idea is developed
further in the following chapter, in a subsequent chapter again featuring
these characters, and finally in the concluding chapter, <q>Einstein's Cat.</q></p>
<p>The quotation that ends this chapter is an <q>aside</q> from Polonius in the
play <cite>Hamlet</cite>, taken from a scene that I have parodied in a
subsequent <q>fit.</q> It seems fitting both for that reason and as my excuse
for what I have written as this <q>fit.</q></p>
<p><i>Recommended resources:</i></p>
<ul>
<li> <a href="http://shakespeare.mit.edu/hamlet/full.html" target="_blank">http://shakespeare.mit.edu/hamlet/full.html</a></li>
</ul>
<h4>Chapter Twenty: Special Relativity</h4>
<p>See Robinson's <cite>Einstein on the Run</cite> regarding Einstein
discussing Poincaré with his mates Habicht and Solovine. The trio
"...argued in detail about a recently published book, <cite>Science and
Hypothesis</cite> . . ."</p>
<p>"As early as 1898, Poincaré questioned the meaning of simultaneity over
large distances and wrote that there may not be a meaningful, universally
applicable time." See Orzel, <cite>Timekeeping</cite>, pp. 199-202 regarding
Poincaré's 1898 <q>The Measure of Time.</q>
<!--Minkowski 1908: see Orzel, Timekeeping, p. 226.--></p>
<p>This bonus figure shows how various proper distances (on the <i>y</i>
axis) become infinite at various speeds (as a percentage of the speed of
light):</p>
<blockquote>
<p><img class="Figure" loading="lazy" src="images/lorentz-graph.png" style="max-height: 500px;"></p>
</blockquote>
<h4>Chapter <i>i</i><sup>2</sup>: Imaginary Geometry</h4>
See <a href="https://en.wikipedia.org/wiki/Minkowski_space#Complex_Minkowski_spacetime"
target="_blank">https://en.wikipedia.org/wiki/<wbr>Minkowski_space#Complex_Minkowski_spacetime</a>
for the quotation below:
<p></p>
<blockquote>In his second relativity paper in 1905-06 Henri Poincaré showed
how, by taking time to be an imaginary fourth spacetime coordinate <i>ict</i>,
where <i>c</i> is the speed of light and <i>i</i> is the imaginary
unit, Lorentz transformations can be visualized as ordinary rotations of
the four dimensional Euclidean sphere <i>x</i><sup>2</sup> + <i>y</i><sup>2</sup>
+ <i>z</i><sup>2</sup> + (<i>ict</i>)<sup>2</sup> = const . Poincaré
set <i>c</i> = 1 for convenience. …<br>
The analogy with Euclidean rotations is only partial since the radius of
the sphere is actually imaginary which turns rotations into rotations in
hyperbolic space (see hyperbolic rotation). This idea, which was mentioned
only briefly by Poincaré, was elaborated by Minkowski in a paper in German
published in 1908 called "The Fundamental Equations for Electromagnetic
Processes in Moving Bodies". Minkowski, using this formulation, restated
the then-recent theory of relativity of Einstein. In particular, by
restating the Maxwell equations as a symmetrical set of equations in the
four variables (<i>x, y, z, ict</i>) combined with redefined vector
variables for electromagnetic quantities, he was able to show directly and
very simply their invariance under Lorentz transformation. He also made
other important contributions and used matrix notation for the first time
in this context. From his reformulation he concluded that time and space
should be treated equally, and so arose his concept of events taking place
in a unified four-dimensional spacetime continuum. </blockquote>
<p>Einstein used both systems of notation in his 1916 general-audience book
<cite>Relativity</cite>.</p>
<h4>Chapter Twenty-Four: Electromagnetism (II)</h4>
<p><i>Recommended resources:</i></p>
<ul>
<li><a href="https://www.youtube.com/watch?v=rB83DpBJQsE" target="_blank">https://www.youtube.com/watch?v=rB83DpBJQsE</a>
Divergence and curl: The language of Maxwell's equations, fluid flow, and more</li>
</ul>
<h4>Chapter Twenty-Five: General Relativity</h4>
<p>I found <a href="https://www.geometrygames.org/CurvedSpaces/" target="_blank">https://www.geometrygames.org/CurvedSpaces/</a>
very amusing. This is NOT the idea of curved spaces that I will be
presenting here, but an idea of this type was entertained by Einstein:
that the universe might be unbounded yet finite. That the
three-dimensional universe could loop back on itself in all directions is
an idea that seems too slippery for my mind to grasp very tightly.</p>
<p>The rows and columns of books of maps is a metaphor for the phase space
explained by Grant Sanderson (<a href="https://www.3blue1brown.com/lessons/differential-equations"
target="_blank">https://www.3blue1brown.com/lessons/differential-equations</a>,
cited above) and Sean Carroll (<cite>Biggest Ideas</cite>, p. 76-80) On
Carroll's page 80: "The entire trajectory is fixed by knowing what point
to start at;" page 99: "According to the Laplacian paradigm, specifying
one point in phase space at one moment in time is enough to determine the
entire trajectory of a system."</p>
<h4>Chapter Twenty-Seven: Quantum Quandaries</h4>
<i>Recommended resources:</i>
<ul>
<li><a href="https://www.3blue1brown.com/lessons/uncertainty-principle" target="_blank">https://www.3blue1brown.com/lessons/uncertainty-principle</a>
<q>The more general uncertainty principle, regarding Fourier transforms</q></li>
<li>The Science Asylum: <a href="https://www.youtube.com/watch?v=LFC2HsT6Bh4"
target="_blank">The True Meaning of Schrödinger's Equation</a></li>
<li>The Science Asylum: <a href="https://www.youtube.com/watch?v=iyN27R7UDnI"
target="_blank">Photons, Entanglement, and the Quantum Eraser</a></li>
<li>The Stanford Encyclopedia of Philosophy: <a href="https://plato.stanford.edu/entries/qt-issues/"
target="_blank">Philosophical Issues in Quantum Theory</a></li>
</ul>
<h4>Fit the Sixth: <i>The Walrus and the Carpenter</i></h4>
<p>"I am he as you are he . . ." In<cite> Rosencrantz and Guildenstern
Are Dead</cite>, the confusion of one character with the other is a
running joke which apparently extends a long-standing thespian tradition:
<a href="https://en.wikipedia.org/wiki/Rosencrantz_and_Guildenstern" target="_blank">https://en.wikipedia.org/<wbr>wiki/<wbr>Rosencrantz_and_Guildenstern</a>.
I'll leave it to you to puzzle out the thematic references I make to
Carroll's poem here.</p>
<h4>Fit the Seventh: <i>Return to the Non Sequitur</i></h4>
<p>Scene one is a satire of the gravediggers scene in <cite>Hamlet</cite> which
takes aim at certain celebrity scientists who shall remain unnamed in this
context. <q>Eric Geller</q> is a mashup of names of two sensationalists of a
bygone generation.</p>
<p>
In this chapter, Hamlet is a quantum system that Rosencrantz intends to
interrogate. In scene two, he is the particle passing through two slits.
Which version of scene two is <q>real</q> depends on its time order relative to
scenes one and three, which take place elsewhere. From Gribbin's <cite>In
Search of Schrödinger's Cat</cite>, page 175: "Our world is a hybrid
combination of the two possible worlds corresponding to the two routes for
the particle, and each world interferes with the other."
</p>
<p>Scene two is parody of another scene in <cite>Hamlet</cite> (Act II, Scene
2). The comedic non sequitur was integral to the original. Some of the
dialogue still fits my story but with a new meaning that touches directly
on the <q>Einstein's Cat</q> question (Introduction and Conclusion).</p>
<p>The limerick Rosencrantz has written on the wall is my adaptation of an
original by A. H. Reginald Buller, published in a 1923 issue of <cite>Punch
</cite> (see <a href="%20https://www.nationalreview.com/corner/a-young-lady-named-bright-etc/"
target="_blank">https://www.nationalreview.com/corner/<wbr>a-young-lady-named-bright-etc/</a>).
This is also a bawdy allusion to Hamlet's mother, who in her <q>haste</q>
<q>wandered</q> in
a <q>relative way</q> in allowing herself to be wooed by her
deceased husband's brother <q>the previous knight</q>,
setting the <q>time</q>
<q>out of joint</q> in a different sense than how the reverse time travel of the
limerick would cause a paradox.</p>
<h4>Chapter Thirty-One: In Defense of a Three-Dimensional Spacetime</h4>
<p>
In the ten years since the publication of <cite>Time Reborn</cite>, Lee Smolin
seems to have found a more satisfying concept of the meaning of the
present: "In the Copenhagen version of quantum mechanics, there is a
quantum world and there is a classical world, and a boundary between them:
when things become definite. When things that are indefinite in the quantum
world become definite. And what they're trying to say is that is the
fundamental thing that happens in nature, when things that are indefinite
become definite. And that's what <q>now</q> is. The moment now, the present
moment, that all these people say is missing from science and missing from
physics, that is the transition from indefinite to definite." <a href="https://bigthink.com/starts-with-a-bang/are-we-approaching-quantum-gravity-all-wrong/"
target="_blank">https://bigthink.com/starts-with-a-bang/are-we-approaching-quantum-gravity-all-wrong/</a>
</p>
<h4>Chapter Thirty-Two: Mapping Spacetime</h4>
<p><i>Recommended resources:</i></p>
<ul>
<li>"<a href="https://www.youtube.com/watch?v=C7tQJ42nGno" target="_blank">Circuit
Energy doesn't FLOW the way you THINK!</a>" This video by Nick Lucid
on <q>The Science Asylum</q> YouTube channel discusses the principle
mentioned at the beginning of the chapter.</li>
</ul>
<h4>Fit The Eighth:<i> The Hall of Mirrors</i></h4>
<p>In this chapter, the <q>Alice</q> we have come to know from the previous
<q>fits</q> turns out to be a story-within-a-story; she is the creation of an
early-mid-90s cyberpunk by the name of Alice who shares an affection for
Lewis Carroll with one of her favorite authors, Douglas Adams. Unlike
Carroll's Alice, this one writes her own story. Inspired in part by the <cite>Hitchhiker's
Guide to the Galaxy</cite>, she has envisioned a future in which
information is more free, not tethered to university enrollment or to a
9600-baud telephone connection. In the words of the Sex Pistols song she
parodied, she is a <q>dog's body</q> with
a <q>council tenancy.</q> In that she has
anticipated such things ahead of her time, she is like Thomasina in
Stoppard's <cite>Arcadia</cite>.</p>
<p>How can you write Alice without parody verse? You have to include it, and
it's best to aim it at something that is taken too seriously. Lewis
Carroll was a brilliant satirist, subverting overly-sober popular poems
and replacing their moral lessons with absurdities ("<a href="https://en.wikipedia.org/wiki/You_Are_Old,_Father_William" target="_blank">You Are Old, Father William</a>"). This was in the same
grand tradition as Warner Brothers studios' take on "Ride of the
Valkyries" starring Bugs Bunny and Elmer Fudd ("<a href="https://en.wikipedia.org/wiki/What%27s_Opera,_Doc%3F" target="_blank">What's Opera, Doc?</a>"), and <q>Weird Al</q> Yankovic's
<q>Theme From Rocky XIII.</q></p>
<p>Chapter one of <cite>Through the Looking-Glass</cite> has one of Alice's two
kittens playing with a ball of yarn and making a mess "all knots and
tangles." Later Tweedledee and Tweedledum spar over a <q>spoiled</q> rattle.</p>
<h4> Conclusion: Einstein's Cat</h4>
<p><i>Recommended resources:</i></p>
<ul>
<li>Arvin Ash: <a href="https://www.youtube.com/watch?v=BZRv8Nko9XQ" target="_blank">Everything
- Yes, Everything - is a SPRING! (Pretty much)</a> </li>
</ul>
<hr>
<h2 class="sticky" id="Bibliography">Bibliography</h2>
<!-- check citations in my essays and paper(s)-->
<h3> Books </h3>
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<li>Bertschinger, Edmund; with Edwin F. Taylor, and John Archibald
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W. W. Norton, 2003.</li>
<li>Wolfson, Richard; and Jay M. Pasachoff. <cite>Physics for Scientists
and Engineers</cite> (third edition). Reading, MA: Addison-Wesley, 1999.</li>
</ul>
<h3> Papers </h3>
<ul>
<li>de Broglie, Louis-Victor. <q>On the Theory of Quanta.</q> 1924.
<a href="https://fondationlouisdebroglie.org/LDB-oeuvres/De_Broglie_Kracklauer.pdf" target="_blank">
https://fondationlouisdebroglie.org/LDB-oeuvres/De_Broglie_Kracklauer.pdf</a>
</li>
<li>Einstein, Albert. <q>On the Electrodynamics of Moving Bodies.</q> 1905.
<a href="https://www.physics.umd.edu/courses/Phys606/spring_2011/einstein_electrodynamics_of_moving_bodies.pdf" target="_blank">https://www.physics.umd.edu/courses/Phys606/spring_2011/einstein_electrodynamics_of_moving_bodies.pdf</a></li>
<li>Riemann, Bernhard. "On the Hypotheses Which Lie at the Bases of
Geometry." <a href="https://www.emis.de/classics/Riemann/WKCGeom.pdf" target="_blank">https://www.emis.de/classics/Riemann/WKCGeom.pdf</a></li>
</ul>
<h3>Articles</h3>
<ul>
<li>Bayley, Melanie. <q>Alice's adventures in algebra: Wonderland solved</q>
<a href="https://www.newscientist.com/article/mg20427391-600-alices-adventures-in-algebra-wonderland-solved/"
target="_blank">https://www.newscientist.com/article/<wbr>mg20427391-600-alices-adventures-in-algebra-wonderland-solved/</a>
</li>
<li>Garret, Ron. <q>Quantum Mysteries Disentangled</q>
<a href="https://flownet.com/ron/QM.pdf"
target="_blank">https://flownet.com/ron/QM.pdf</a> See also <a href="https://www.youtube.com/watch?v=dEaecUuEqfc" target="_blank">https://www.youtube.com/watch?v=dEaecUuEqfc</a></li>
<li>Honner, Patrick. <q>The (Imaginary) Numbers at the Edge of Reality</q>
<a href="https://www.quantamagazine.org/the-imaginary-numbers-at-the-edge-of-reality-20181025/" target="_blank">https://www.quantamagazine.org/<wbr>the-imaginary-numbers-at-the-edge-of-reality-20181025/</a>
</li>
<li>Orzel, Chad. <q>How Do You Create Quantum Entanglement?</q>
<a href="https://www.forbes.com/sites/chadorzel/2017/02/28/how-do-you-create-quantum-entanglement/"
target="_blank">https://www.forbes.com/sites/chadorzel/2017/02/28/<wbr>how-do-you-create-quantum-entanglement/</a></li>
<li>Russ, Kelley. <q>The Ontology and Cosmology of Non-Euclidean Geometry</q> <a
href="http://www.friesian.com/curved-1.htm" target="_blank">http://www.friesian.com/curved-1.htm</a>
</li>
<li>Siegel, Ethan. <q>Are we approaching gravity all wrong?</q>
<a href="https://bigthink.com/starts-with-a-bang/are-we-approaching-quantum-gravity-all-wrong/" target="_blank">https://bigthink.com/starts-with-a-bang/<wbr>are-we-approaching-quantum-gravity-all-wrong/</a></li>
<li>Siegel, Ethan. <q>Does Time Really Exist?</q>
<a href="https://bigthink.com/starts-with-a-bang/does-time-exist-182965/" target="_blank">https://bigthink.com/starts-with-a-bang/<wbr>does-time-exist-182965/</a></li>
<li>Thomson, Jonny. <q>A brief history of (linear) time</q>
<a href="https://bigthink.com/thinking/a-brief-history-of-linear-time/" target="_blank">https://bigthink.com/thinking/a-brief-history-of-linear-time/</a></li>
<li>Wood, Charlie. <q>The Strange Numbers That Birthed Modern Algebra</q>
<a href="https://www.quantamagazine.org/the-strange-numbers-that-birthed-modern-algebra-20180906/" target="_blank">https://www.quantamagazine.org/the-strange-numbers-that-birthed-modern-algebra-20180906/</a></li>
<li>Wolchover, Natalie. <q>The Peculiar Math That Could Underlie the Laws of Nature</q>
<a href="https://www.quantamagazine.org/the-octonion-math-that-could-underpin-physics-20180720/" target="_blank">https://www.quantamagazine.org/<wbr>the-octonion-math-that-could-underpin-physics-20180720/</a></li>
</ul>
<h3> Videos </h3>
<ul>
<li>3blue1brown: <a href="https://www.3blue1brown.com/lessons/abstract-vector-spaces" target="_blank">Abstract vector spaces</a> </li>
<li>3blue1brown: <a href="https://www.3blue1brown.com/lessons/determinant" target="_blank">The determinant</a></li>
<li>3blue1brown: <a href="https://www.3blue1brown.com/lessons/differential-equations" target="_blank">Differential equations, studying the unsolvable</a></li>
<li>3blue1brown: <a href="https://www.youtube.com/watch?v=rB83DpBJQsE" target="_blank">Divergence
and curl: The language of Maxwell's equations, fluid flow, and more</a></li>
<li>3blue1brown: <a href="https://www.3blue1brown.com/lessons/dot-products" target="_blank">Dot products</a></li>
<li>3blue1brown: <a href="https://www.3blue1brown.com/lessons/eulers-formula-via-group-theory" target="_blank">Euler's formula with introductory group theory</a></li>
<li>3blue1brown: <a href="https://www.3blue1brown.com/lessons/groups-and-monsters
target="_blank">Group theory, abstraction, and the 196,883-dimensional
monster</a></li>
<li>3blue1brown: <a href="https://www.3blue1brown.com/lessons/inverse-matrices" target="_blank">Inverse matrices, column space and null space</a></li>
<li>3blue1brown: <a href="https://www.3blue1brown.com/lessons/span" target="_blank">Linear
combinations, span, and basis vectors</a></li>
<li>3blue1brown: <a href="https://www.3blue1brown.com/lessons/linear-transformations"
target="_blank">Linear transformations and matrices</a> </li>
<li>3blue1brown: <a href="https://www.3blue1brown.com/lessons/uncertainty-principle" target="_blank"> The more general uncertainty principle, regarding Fourier transforms</a> </li>
<li>Arvin Ash: <a href="https://www.youtube.com/watch?v=BZRv8Nko9XQ" target="_blank">Everything
- Yes, Everything - is a SPRING! (Pretty much)</a></li>
<li>Dialect: <a href="https://www.youtube.com/watch?v=Dn0ZZRVuJcU" target="_blank">The
meaning of the metric tensor</a></li>
<li>DrPhysicsA: <a href="https://www.youtube.com/watch?v=U685DR19XyY&list=PLDB15F7E29A5F0426"
target="_blank">Special and General Relativity</a> playlist</li>
<li>DrPhysicsA: <a href="https://www.youtube.com/watch?v=IsX5iUKNT2k&list=PL04722FAFB07E38E1"
target="_blank">Quantum Mechanics</a> playlist</li>
<li>eigenchris: <a href="https://www.youtube.com/playlist?list=PLJHszsWbB6hrkmmq57lX8BV-o-YIOFsiG"
target="_blank">Tensors for Beginners</a> playlist </li>
<li>eigenchris: <a href="https://www.youtube.com/watch?v=kGXr1SF3WmA&list=PLJHszsWbB6hpk5h8lSfBkVrpjsqvUGTCx"
target="_blank">Tensor Calculus</a> playlist </li>
<li>eigenchris: <a href="https://www.youtube.com/watch?v=kGXr1SF3WmA&list=PLJHszsWbB6hpk5h8lSfBkVrpjsqvUGTCx"
target="_blank">Relativity</a> playlist </li>
<li>MinutePhysics: <a href="https://www.youtube.com/watch?v=zcqZHYo7ONs">Bell's
Theorem: The Quantum Venn Diagram Paradox </a></li>
<li>MIT OpenCourseWare Physics 8.01 (Physics I): <a href="https://www.youtube.com/watch?v=wWnfJ0-xXRE&list=PLyQSN7X0ro203puVhQsmCj9qhlFQ-As8e"
target="_blank">Classical Mechanics</a> </li>
<li>MIT OpenCourseWare Physics 8.02 (Physics II): <a href="https://www.youtube.com/watch?v=rtlJoXxlSFE&list=PLyQSN7X0ro2314mKyUiOILaOC2hk6Pc3j"
target="_blank">Electricity and Magnetism</a></li>
<li>MIT OpenCourseWare Physics 8.03 (Physics III): <a href="https://www.youtube.com/watch?v=DUYxVwXZbCU&list=PLUdYlQf0_sSsdOhQ_8jfrAGzbGbJ7MXGe"
target="_blank">Vibrations and Waves</a></li>
<li>MIT OpenCourseWare Physics 8.04 (<a href="https://www.youtube.com/watch?v=lZ3bPUKo5zc&list=PLUl4u3cNGP61-9PEhRognw5vryrSEVLPr"
target="_blank">Quantum Physics I</a>)</li>
<li>MIT OpenCourseWare Physics 8.05 (<a href="https://www.youtube.com/watch?v=QI13S04w8dM&list=PLUl4u3cNGP60QlYNsy52fctVBOlk-4lYx"
target="_blank">Quantum Physics II</a>)</li>
<li>MIT OpenCourseWare: Physics 8.224 <a href="https://ocw.mit.edu/courses/8-224-exploring-black-holes-general-relativity-astrophysics-spring-2003/video_galleries/video-lectures/"
target="_blank">Exploring Black Holes</a></li>
<li>Physics with Elliot: <a href="https://www.youtube.com/watch?v=W8QZ-yxebFA"
target="_blank">To Understand the Fourier Transform, Start From
Quantum Mechanics</a></li>
<li>The Science Asylum: <a href="https://www.youtube.com/watch?v=C7tQJ42nGno"
target="_blank">Circuit Energy doesn't FLOW the way you THINK!</a></li>
<li>The Science Asylum: <a href="https://www.youtube.com/watch?v=iyN27R7UDnI"
target="_blank">Photons, Entanglement, and the Quantum Eraser</a><br>
</li>
<li>The Science Asylum: <a href="https://www.youtube.com/watch?v=LFC2HsT6Bh4"
target="_blank">The True Meaning of Schrödinger's Equation</a></li>
<li>Veritasium: <a href="https://www.youtube.com/watch?v=bHIhgxav9LY" target="_blank">The
Big Misconception About Electricity</a></li>
<li>Veritasium: <a href="https://www.youtube.com/watch?v=cUzklzVXJwo" target="_blank">How
Imaginary Numbers Were Invented</a> </li>
</ul>
<hr>
<h2 class="sticky" id="AboutTheAuthor">About the Author</h2>
<p>Kent Heiner lives in northwest Washington state. He studied math, physics
and computer science at Western Washington University and has an
undergraduate degree in international relations from Brigham Young
University. He is also the author of <cite>Without Smoking Gun:
Investigating the Death of LCDR William B. Pitzer</cite> (TrineDay,
2004). He recently concluded a twenty-year career as a database
administrator in the insurance and telecommunications industries and will
be looking for something different after the completion of this book. Kent
maintains a website at <a href="https://www.monstro.us">www.monstro.us</a>
and welcomes your feedback. If you find some value in this book, you can
make a donation at <a href="https://ko-fi.com/heiner" target="_blank">ko-fi.com/heiner</a>.</p>
<hr>
<h2 class="sticky" id="Reviews">Hype</h2>
<p>Is it a mashup of absurdist fiction? A physics study guide? Why would any sane person want to read this?</p>
<p><q>Pundits and gurus outgrowing their britches? Reductive satire that keeps
you in stitches.</q> — Maria von T. </p>
<p><q>The long-awaited completion of the <cite>Hamlet</cite> trilogy which also
manages somehow to incorporate a natural completion of the story arc begun
in <cite>Alice's Adventure in Wonderland</cite>. A must-read for lovers of
any kind of book whatsoever.</q> — No one, ever. Or Henri Poincaré,
eventually?</p>
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